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<H2><A Name="MB04RD">MB04RD</A></H2>
<H3>
Reduction of a real matrix pencil in generalized real Schur form to a block-diagonal form
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To reduce a real matrix pair (A,B) in generalized real Schur form
to a block-diagonal form using well-conditioned non-orthogonal
equivalence transformations. The condition numbers of the left and
right transformations used for the reduction are roughly bounded
by PMAX, where PMAX is a given value. The transformations are
optionally postmultiplied in the given matrices X and Y. The
generalized Schur form is optionally ordered, so that clustered
eigenvalues are grouped in the same pair of blocks.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE MB04RD( JOBX, JOBY, SORT, N, PMAX, A, LDA, B, LDB, X,
$ LDX, Y, LDY, NBLCKS, BLSIZE, ALPHAR, ALPHAI,
$ BETA, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBX, JOBY, SORT
INTEGER INFO, LDA, LDB, LDWORK, LDX, LDY, N, NBLCKS
DOUBLE PRECISION PMAX, TOL
C .. Array Arguments ..
INTEGER BLSIZE(*), IWORK(*)
DOUBLE PRECISION A(LDA,*), ALPHAI(*), ALPHAR(*), B(LDB,*),
$ BETA(*), DWORK(*), X(LDX,*), Y(LDY,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
JOBX CHARACTER*1
Specifies whether or not the left transformations are
accumulated, as follows:
= 'N': The left transformations are not accumulated;
= 'U': The left transformations are accumulated in X
(the given matrix X is updated).
JOBY CHARACTER*1
Specifies whether or not the right transformations are
accumulated, as follows:
= 'N': The right transformations are not accumulated;
= 'U': The right transformations are accumulated in Y
(the given matrix Y is updated).
SORT CHARACTER*1
Specifies whether or not the diagonal blocks of the
generalized real Schur form are reordered, as follows:
= 'N': The diagonal blocks are not reordered;
= 'S': The diagonal blocks are reordered before each
step of reduction, so that clustered eigenvalues
appear in the same pair of blocks.
= 'C': The diagonal blocks are not reordered, but the
"closest-neighbour" strategy is used instead of
the standard "closest to the mean" strategy (see
METHOD);
= 'B': The diagonal blocks are reordered before each
step of reduction, and the "closest-neighbour"
strategy is used (see METHOD).
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the matrices A, B, X and Y. N >= 0.
PMAX (input) DOUBLE PRECISION
An upper bound for the absolute value of the elements of
the individual transformations used for reduction
(see METHOD and FURTHER COMMENTS). PMAX >= 1.0D0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N upper quasi-triangular part
of this array must contain the upper quasi-triangular
matrix A in the generalized real Schur form, as returned
by the LAPACK Library routine DGGES. The lower triangular
part below the Schur matrix is used as workspace.
On exit, the leading N-by-N upper quasi-triangular part of
this array contains the computed block-diagonal matrix, in
real Schur canonical form, corresponding to the given
matrix A. The remaining part is set to zero.
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB,N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix B in the
generalized real Schur form, as returned by the LAPACK
Library routine DGGES. The diagonal elements of B are
non-negative. The strictly lower triangular part is used
as workspace. The matrix B is assumed nonzero.
On exit, the leading N-by-N upper triangular part of this
array contains the computed upper triangular block-
diagonal matrix, corresponding to the given matrix B. The
remaining part is set to zero. The diagonal elements of B
are non-negative.
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,N).
X (input/output) DOUBLE PRECISION array, dimension (LDX,*)
On entry, if JOBX = 'U', the leading N-by-N part of this
array must contain a given matrix X, for instance the left
transformation matrix VSL returned by the LAPACK Library
routine DGGES.
On exit, if JOBX = 'U', the leading N-by-N part of this
array contains the product of the given matrix X and the
left transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-orthogonal equivalence transformations
having elements with magnitude less than or equal to PMAX.
If JOBX = 'N', this array is not referenced.
LDX INTEGER
The leading dimension of the array X.
LDX >= 1, if JOBX = 'N';
LDX >= MAX(1,N), if JOBX = 'U'.
Y (input/output) DOUBLE PRECISION array, dimension (LDY,*)
On entry, if JOBY = 'U', the leading N-by-N part of this
array must contain a given matrix Y, for instance the
right transformation matrix VSR returned by the LAPACK
Library routine DGGES.
On exit, if JOBY = 'U', the leading N-by-N part of this
array contains the product of the given matrix Y and the
right transformation matrix that reduced (A,B) to block-
diagonal form. The local transformation matrix is itself a
product of non-orthogonal equivalence transformations
having elements with magnitude less than or equal to PMAX.
If JOBY = 'N', this array is not referenced.
LDY INTEGER
The leading dimension of the array Y.
LDY >= 1, if JOBY = 'N';
LDY >= MAX(1,N), if JOBY = 'U'.
NBLCKS (output) INTEGER
The number of diagonal blocks of the matrices A and B.
BLSIZE (output) INTEGER array, dimension (N)
The first NBLCKS elements of this array contain the orders
of the resulting diagonal blocks of the matrices A and B.
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N)
BETA (output) DOUBLE PRECISION array, dimension (N)
On exit, if INFO = 0, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j),
j = 1, ..., N, will be the generalized eigenvalues.
ALPHAR(j) + ALPHAI(j)*i, and BETA(j), j = 1, ..., N, are
the diagonals of the complex Schur form (S,T) that would
result if the 2-by-2 diagonal blocks of the real Schur
form of (A,B) were further reduced to triangular form
using 2-by-2 complex unitary transformations.
If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with ALPHAI(j+1) negative.
All BETA(j) are non-negative real numbers.
The quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
easily over- or underflow, and BETA(j) may even be zero.
Thus, the user should avoid naively computing the ratio.
If A and B are obtained from general matrices using DGGES,
ALPHA will be always less than and usually comparable with
norm(A) in magnitude, and BETA always less than and
usually comparable with norm(B).
</PRE>
<B>Tolerances</B>
<PRE>
TOL DOUBLE PRECISION
If SORT = 'S' or 'B', the tolerance to be used in the
ordering of the diagonal blocks of the upper triangular
matrix pair.
If the user sets TOL > 0, then the given value of TOL is
used as an absolute tolerance: a pair of blocks i and a
temporarily fixed pair of blocks 1 (the first pair of
blocks of the current trailing pair of submatrices to be
reduced) are considered to belong to the same cluster if
their eigenvalues satisfy the following "distance"
condition
| lambda_1 - lambda_i | <= TOL.
If the user sets TOL < 0, then the given value of TOL is
used as a relative tolerance: a pair of blocks i and a
temporarily fixed pair of blocks 1 are considered to
belong to the same cluster if their eigenvalues satisfy,
for finite lambda_j, j = 1, ..., N,
| lambda_1 - lambda_i | <= | TOL | * max | lambda_j |.
If the user sets TOL = 0, then an implicitly computed,
default tolerance, defined by TOL = SQRT( SQRT( EPS ) )
is used instead, as a relative tolerance, where EPS is
the machine precision (see LAPACK Library routine DLAMCH).
The approximate symmetric chordal metric is used as
"distance" of two complex, possibly infinite numbers, x
and y. This metric is given by the formula
d(x,y) = min( |x-y|, |1/x-1/y| ),
taking into account the special cases of infinite or NaN
values.
If SORT = 'N' or 'C', this parameter is not referenced.
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension (N+6)
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
On exit, if INFO = -21, DWORK(1) returns the minimum
value of LDWORK. When LDWORK = 0 is set on entry, the
routine will return this value for INFO, and also set
DWORK(1), but no error message related to LDWORK is
issued by XERBLA.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 1, if N <= 1;
LDWORK >= 4*N + 16, if N > 1.
If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the
DWORK array, returns this value as the first entry of
the DWORK array, and no error message related to LDWORK
is issued by XERBLA.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the matrix pencil defined by A and B is singular.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
Consider first that SORT = 'N'. Let
( A A ) ( B B )
( 11 12 ) ( 11 12 )
A = ( ), B = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
be the given matrix pair in generalized real Schur form, where
initially A and B are the first pair of diagonal blocks of
11 11
dimension 1-by-1 or 2-by-2. An attempt is made to compute the
transformation matrices X and Y of the form
( I V ) ( I W )
X = ( ), Y = ( ) (1)
( 0 I ) ( 0 I )
(partitioned as A and B ), so that (' denotes the transpose)
( A 0 ) ( B 0 )
( 11 ) ( 11 )
X' A Y = ( ), X' B Y = ( ),
( 0 A ) ( 0 B )
( 22 ) ( 22 )
and the elements of V and W do not exceed the value PMAX in
magnitude. An adaptation of the standard method for solving
generalized Sylvester equations [1], which controls the magnitude
of the individual elements of the computed solution [2], is used
to obtain V and W. When this attempt fails, a 1-by-1 (or 2-by-2)
pair of diagonal blocks of (A , B ), whose eigenvalue(s) is
22 22
(are) the closest to the mean of those of (A , B ) is selected,
11 11
and moved by orthogonal equivalence transformations in the leading
position of (A , B ); the moved diagonal blocks in A and B are
22 22
then added to the blocks A and B , respectively, increasing
11 11
their order by 1 (or 2). Another attempt is made to compute
suitable transformation matrices X and Y with the new definitions
of the blocks A , A , B , and B . After successful
11 22 11 22
transformation matrices X and Y have been obtained, they
postmultiply the current transformation matrices (if JOBX = 'U'
and/or JOBY = 'U') and the whole procedure is repeated for the new
blocks A and B .
22 22
When SORT = 'S', the diagonal blocks of the generalized real Schur
form are reordered before each step of the reduction, so that each
cluster of generalized eigenvalues, defined as specified in the
definition of TOL, appears in adjacent blocks. The blocks for
each cluster are merged together, and the procedure described
above is applied to the larger blocks. Using the option SORT = 'S'
will usually provide better efficiency than the standard option
(SORT = 'N'), proposed in [2], because there could be no or few
unsuccessful attempts to compute individual transformation
matrices X and Y of the form (1). However, the resulting
dimensions of the blocks are usually larger; this could make
subsequent calculations less efficient.
When SORT = 'C' or 'B', the procedure is similar to that for
SORT = 'N' or 'S', respectively, but the blocks of A and B
22 22
whose eigenvalue(s) is (are) the closest to those of (A , B )
11 11
(not to their mean) are selected and moved to the leading position
of A and B . This is called the "closest-neighbour" strategy.
22 22
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[1] Kagstrom, B. and Westin, L.
Generalized Schur Methods with Condition Estimators for
Solving the Generalized Sylvester Equation.
IEEE Trans. Auto. Contr., 34, pp. 745-751, 1989.
[2] Bavely, C. and Stewart, G.W.
An Algorithm for Computing Reducing Subspaces by Block
Diagonalization.
SIAM J. Numer. Anal., 16, pp. 359-367, 1979.
[3] Demmel, J.
The Condition Number of Equivalence Transformations that
Block Diagonalize Matrix Pencils.
SIAM J. Numer. Anal., 20, pp. 599-610, 1983.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3 4
The algorithm usually requires 0(N ) operations, but 0(N ) are
possible in the worst case, when the matrix pencil cannot be
diagonalized by well-conditioned transformations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
The individual non-orthogonal transformation matrices used in the
reduction of A and B to a block-diagonal form have condition
numbers of the order PMAX. This does not guarantee that their
product is well-conditioned enough. The routine can be easily
modified to provide estimates for the condition numbers of the
clusters of generalized eigenvalues.
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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